Cluster Groups

WS 2019/2020

Quantum Mechanics for Mathematicians

  • Newtonian Classical Mechanics (affine spaces, Galilean transformations, Newton's equation)
  • Newtonian Classical Mechanics (examples, kinetic and potential energy)
  • Lagrangian Classical Mechanics (motivation, calculus of variations)
  • Lagrangian Classical Mechanics (Lagrangian function and action, Euler-Lagrange formula, Hamilton's equation)
  • Observables (algebra of observables, Poisson-bracket, analogies to quantum mechanics)
  • Axioms of Quantum Mechanics (states, observables, measurement, correspondence principle, position and momentum operators)
  • Schrödinger Equation (wavefunction, Hamilton operator, eigenvalue equation and eigenstates)
  • Schrödinger Equation continued (eigenstates for potential well and potential step), Double-Slit Experiment (interference, probability amplitude calculus)

Riemannian geometry and Brownian motion

  • Tangent Vectors, The Differential of a Smooth Map, The Tangent Bundle (reference: “Introduction to Smooth Manifolds” by John M. Lee)
  • Connection, Vector Fields, Covariant Derivative, Geodesic and Parallel Translation
  • Discussion about curvature of space and Gauss-Bonnet Theorem
  • Divergence theorem, Green's identities
  • Laplace–Beltrami operator
  • Lie group and principal bundle

SS 2019

Differential Geometry

  • differentiable structures, manifolds, tangent spaces
  • de Rham cohomology, embeddings of manifolds, regular value theorem
  • (pseudo-)Riemannian manifolds, Riemannian measures, Laplace-Beltrami operator, Gauss-Green identity, Stokes' theorem
  • Covariant derivative, Riemannian connection, Christoffel symbols, Parallel displacement, exponential mapping (cf. Section 5D, Kuehnel: Differential geometry)
  • Curvature tensor and intrinsic geometry of surfaces: Equations of Gauss and Weingarten, Theorema egregium, Fundamental theorem of the local theory of surfaces (Sections 4A-D, Kuehnel)
  • More on the curvature tensor and Weingarten map, Sectional curvature, Ricci tensor and curvature, Einstein tensor (Chapters 4,6: Kuehnel)
  • Spaces of constant curvature: Hyperbolic space, Geodesics and Jacobi fields, Local isometry of spaces of constant curvature (Chapter 7: Kuehnel)


WS 2018/2019

Analysis and Probability

  • Organizational meeting
  • Probability that random polynomial has no real roots (Talk by Anna Gusakova)
  • Theorems on positive real functions (Talk by Anna Muranova)
  • Non-explosion of solutions to SDEs with singular coefficients (Talk by Chengcheng Ling)
  • Emergence of Flocking in the Fractional Cucker-Smale Model (Talk by Peter Kuchling)
  • Robust Poincaré inequality and Application to the transitional phase of fractional Laplacian (Talk by Guy Fabrice Foghem Gounoue)
  • My problems and failed ideas regarding the edge fluctuation of non hermitian random matrices with independent entries (Jonas Jalowy)
  • Effective impedance of a finite electric network. Definitions and main properties (theorems and conjectures) (Anna Muranova)
  • Application of Probability Methods in Number Theory and Integral Geometry (Anna Gusakova)
  • Ordered fields. The ordered field of rational functions“ (Anna Muranova)
  • Dynamics on the cone: an overview of my thesis (Peter Kuchling)
  • Well-posedness of SDE driven by Levy noise (Chengcheng Ling)
  • Discussion about the talks for workshop/retreat

Statistical Mechanics

  • Point processes 3: Moment Problem, Local Convergence Jansen, Gibbsian Point Processes, pp. 33 - 45
  • Point processes 2: Intensity Measure, Correlation Functions, Generating Functionals Jansen, Gibbsian Point Processes, pp. 25 - 33
  • Point processes 1: Configuration Spaces, Observables, PPP Jansen, Gibbsian Point Processes, pp. 15 - 25
  • Introduction to partition functions
  • Partition Functions and Gibbs Measures
  • Correlation Functions
  • Phase Transitions (Critical Exponents and Models)
  • Scaling Limits
  • Mean Field Limit of the Cucker-Smale model: Two Approaches (Peter Kuchling)

WS 2017/2018

Graph Theory

Reading Spectral graph theory by Fan R. K. Chung

  • Chapter 2: Isoperimetric problems
  • Chapter 3: Diameters and eigenvalues
  • Talk by Anna Muranova: On effective resistance of electric network with impedances
  • Talk by Melissa Meinert: Ollivier Ricci curvature for general Laplacians
  • Markov chains
  • Talk by Filip Bosnic: Examples of Poincaré and log-Sobolev inequalities on discrete configuration spaces
  • Introduction to random graphs by Alan Frieze and Michał Karoński

Lévy processes

Discuss Lévy Processes and Infinitely Divisible Distributions by Ken-iti Sato

  • Chapter 1 (pp. 1 - 30): Examples of Levy Processes
  • Chapter 2 (pp. 31 - 68): Characterisation and Existence of Levy Processes and Additive Processes
  • Chapter 2 (pp. 31 - 41): Infinitely Divisible Processes and the Lévy-Khintchine formula
  • Chapter 2 (pp. 54 - 67): Transition Functions and the Markov Property; Existence of Lévy and Additive Processes
  • Chapter 3 (pp. 69 - 77): Selfsimilar and Semi-selfsimilar Processes and their Exponents
  • Chapter 3 (pp. 77 - 90): Representations of Stable and Semi-stable Distributions
  • Viswanathan et al. Papers: “Levy Flight Search Patterns of Wandering Albatrosses” and “Optimizing the Success of Random Searches”
  • Bertoin (pp. 18-24): Potential theory: Markov property and important definitions
  • Bertoin (pp. 43-48): Duality and time reversal
  • Bertoin (pp. 48-56): Potential theory: capacity measure; essentially polar sets and capacity
  • Bertoin (pp. 56-68): Potential theory: energy; the case of a single point
  • Bertoin (pp. 71-84): Subordinators: definitions; passage across a level; arcsine laws
  • Bertoin (pp. 84-99): Subordinators: rates of growth; (Hausdorff-)dimension of the range

Analysis

Discuss

  • Maximum principle for ecliptic operators (Gibarg-Trudinger)
  • Paley-Littlewood Decomposition (Grafakos classical Fourrier Analysis, second or third edtion)
  • Another looks at Sobolev spaces (Articles Brezis Bourgain_Mironescu)
  • Spectral theory: spectral measure for Unbounded operators
  • Spectral theory for unbounded operator and spectral measure (Book by Reed-Simon)
  • Restriction of the Fourier Transform on the sphere (Book by Elias Stein)
  • Interplation of Banach spaces and application (Lecture note by Alessandra Lunadri Theory of interpolation 2009)
  • Introduction to optimal transport theory (Book by Cedri Villani: Old and New, it is free on-line)
  • Introduction to control and non linearity (Book by Jean-Michel Coron: Control and Non-linearity free one his we-page)
  • Introduction to Calculus of variations

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